> Posted by: "Jorge Petrúcio Viana" petrucio@...

> Fri Jul 13, 2007 4:23 am (PST)

>

> To me, seems that algebraic, order and topological BAs are closer to

> each other than to logical BAs.

Initially I was sceptical that these constituted distinct notions---my

reaction was, aren't algebraic, order, and logical BAs all the same?

(Topological BAs are different of course, being those BAs with a

topology and continuous homomorphisms.)

However if one views an order BA as a poset with the requisite sups and

infs satisfying the requisite laws, taking the signature to be that of a

poset with no lattice operations, one could reasonably argue that the

morphisms should be just all monotone functions. Is there any other

notion of order BA that gives it a different extension from the

algebraic or ordinary BAs?

In the same vein, one could reasonably take the logical BAs to be either

the initial BA (the prototypical BA defining the equational theory) or

the free BAs (although one might question the value to logic of the

uncountably generated free BAs).

The method in the apparent madness of the different rules for

topological and order BAs (morphisms respecting the meets and joins of

the former but not the latter) can be illustrated with the example of

CABAs, the complete atomic BAs. Since CABAs are automatically

topological BAs, the above reasoning distinguishing order BAs from

ordinary BAs when applied to CABAs should result in topological CABAs

being distinguished from ordinary CABAs by taking their signature to

exclude the sups and infs and to contain only the topology and the order

(the latter being subsumable by the topology only if one gives up the

convention that a CABA is a totally order disconnected space) and hence

the morphisms to be the continuous monotone functions. Topological

CABAs follow a different rule than topological BAs for the same reason

order BAs follow a different rule than ordinary BAs: if they followed

the same rule topological CABAs would merely be CABAs, just as order BAs

would merely be BAs, and hence a waste of a potentially useful term.

None of this however seems to support Jorge's sense that the algebraic,

order, and topological BAs form a natural cluster separate from the

logical BAs. On the other hand one can at least make the argument that,

among all adjunctions F |- U for which UF is the monad (equational

theory) of Boolean algebras, the logical and ordinary BAs are at the

distal end from Set of the initial and final such adjunctions, being

respectively the Kleisli and Eilenberg-Moore categories. In that sense

they are as far apart as they can get. Absent an equally compelling

force separating order and topological BAs from ordinary BAs, perhaps

the natural gravity of mathematics pulls these three bodies into a tight

but chaotic orbit around each other.

We now know by hard experience to check the radars of journals. Lattice

theory being completely off that of the Journal of Irreproducible

Results, it would be irresponsible to waste its referees' time by

submission of any substantive development of the above theme. As Arthur

C. Clarke would have put it, any sufficiently advanced mathematics is

indistinguishable from gibberish.

Vaughan Pratt